# Two students studied three theorems of four theories

Ηλεκτρονική - Συσκευές

8 Οκτ 2013 (πριν από 4 χρόνια και 9 μήνες)

102 εμφανίσεις

2-3-4
24/08/2004

THE THIRTEEN INTERPRETATIONS OF THE SENTENCE
Two students studied three theorems of four theories

There are two students (Mary and Bill) each of which studied the same three theorems (Th1, Th2 and
Th3), all of which belong to a well-defined group of four theories (Tr1, Tr2, Tr3, Tr4).
s
tud
y

1
CONTEXT
f
ou
r

three
two

Th1
Th2
Th3
Tr1
Tr2
Tr3
Tr4
s
tudy
'

x1

x2

y1

y2

y3

z1

z2

z3

z4

M
ar
y

[x1≠x2 ∧ y1≠y2≠y3 ∧ z1≠z2≠z3≠z4 ∧

x
[(x = x1 ∨ x = x2) → student’(x)] ∧
B
il
l

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) → theory’(z)] ∧

y

z
[((y=y1 ∨ y=y2 ∨ y=y3) ∧ (z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) →
(theorem’(y) ∧ belong’(y, z))] ∧
0 0 0

x

y
[((x=x1 ∨ x=x2) ∧ (y=y1 ∨ y=y2 ∨ y=y3)) → study’(x,y)]]

belong

There are two students (Mary and Bill) each of which studied the same three theorems (Th1, Th2 and
Th3), each of which belong to a group of four theories, but the theories may possibly be different for
the different teorems (Tr11, Tr12, Tr13, Tr14, Tr21, Tr22, Tr23, Tr24, Tr31, Tr32, Tr33, Tr34).
2
CONTEXT
s
tud
y

f
ou
r

three
two

Th1
Th2
Th3
Tr21
Tr22
Tr23
Tr24
Tr31
Tr32
Tr33
Tr34
Tr11
Tr12
Tr13
Tr14

x1

x2

y1

y2

y3
[x1≠x2 ∧ y1≠y2≠y3 ∧

x
[(x = x1 ∨ x = x2) → student’(x)] ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) →
(theorem’(y) ∧
M
ar
y

z1

z2

z3

z4
[z1≠z2≠z3≠z4 ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) → theory’(z)] ∧
0 0 01
B
il
l

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) →
belong’(y,z)]])] ∧

x

y
[((x=x1 ∨ x=x2) ∧ (y=y1 ∨ y=y2 ∨ y=y3)) →
study’(x, y)]]

s
tud
y

3
CONTEXT
f
ou
r

three
two
There are two students (Mary and Bill) and a well-defined group of four theories (Tr1, Tr2, Tr3, Tr4).
Each student chooses three theorems belonging to all four theories, but they may choose different
theorems

x1

x2

z1

z2

z3

z4
[z1≠z2≠z3≠z4 ∧ x1≠x2 ∧
ThM1
ThM2
ThM3
ThB3
ThB2
ThB1
Tr4
Tr3
Tr2
Tr1

x
[(x = x1 ∨ x = x2) → student’(x)] ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) → theory’(z)] ∧
M
ar
y

x
[(x=x1 ∨ x=x2)) →

y1

y2

y3
[y1≠y2≠y3 ∧
B
il
l

y
[(y=y1 ∨ y=y2 ∨ y=y3) →
0 00 0
(theorem’(y) ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) →
belong’(y, z)])] ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) → study’(x, y)]]]]

There are two students (Mary and Bill). Each student chooses three theorems (ThM1, ThM2, ThM3,
ThB1, ThB2, ThB3) and four theories (TrM1, TrM2, TrM3, TrM4, TrB1, TrB2, TrB3, TrB4). Each
theorem chosen by a student must belong to all the theories chosen by her/him.
CONTEXT
4
s
tud
y

f
ou
r

three
two

x1

x2
[x1≠x2 ∧
ThM1
ThM2
ThM3
TrB4
TrB3
TrB2
TrB1
ThB3
ThB2
ThB1
TrM4
TrM3
TrM2
TrM1

x
[(x = x1 ∨ x = x2) → student’(x)] ∧

x
[(x = x1 ∨ x = x2) →
M
ar
y

y1

y2

y3

z1

z2

z3

z4
[y1≠y2≠y3 ∧ z1≠z2≠z3≠z4 ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) → theory’(z)] ∧

y

z
[((y=y1 ∨ y=y2 ∨ y=y3) ∧
0 00 00
B
il
l

(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4)) →
(theorem’(y) ∧ belong’(y, z))] ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) → study’(x, y)]]]]

1
2-3-4
24/08/2004

There are two students (Mary and Bill). Each student chooses three theorems (ThM1, ThM2, ThM3,
ThB1, ThB2, ThB3) each of which must belong to any four theories (TrM11, TrM12, TrM13, TrM14,
for theorem ThM1; TrM21, TrM22, TrM23, TrM24, for theorem ThM2 and so on).
5
CONTEXT

ThM1
ThM3
ThB1
ThB2
TrB34
TrB33
TrB32
TrB31
ThB3
TrM24
TrM23
TrM22
TrM21
ThM2
TrM14
TrM13
TrM12
TrM11
s
tud
y

f
ou
r

three
two

x1

x2
[ x1≠x2 ∧

x
[(x = x1 ∨ x = x2) → student’(x)] ∧

x
[(x = x1 ∨ x = x2) →

y1

y2

y3
[y1≠y2≠y3 ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) →
(theorem’(y) ∧
M
ar
y

z1

z2

z3

z4
[z1≠z2≠z3≠z4 ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) →
0 00 01
theory’(z)] ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) →
belong’(y, z)]])] ∧
B
il
l

y
[(y=y1 ∨ y=y2 ∨ y=y3) → study’(x, y)]]]]

There are two students (Mary and Bill), four theories (Tr1, Tr2, Tr3, Tr4), and three theorems for
each theory. Each student studies those theorems

s
tud
y

6
CONTEXT
f
ou
r

three
two
Th21
Th22
Th23
M
ar
y

Th43
Th42
Th41
Th13
Th12
Th11
Th33
Th32
Th31
Tr4
Tr3
Tr2
Tr1
B
il
l

x1

x2

z1

z2

z3

z4
[x1≠x2 ∧ z1≠z2≠z3≠z4 ∧

x
[(x = x1 ∨ x = x2) → student’(x)] ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) → theory’(z)] ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) →

y1

y2

y3
[y1≠y2≠y3 ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) →
(theorem’(y) ∧ belong’(y,z))] ∧
0 010 0

x

y
[((x=x1 ∨ x= x2) ∧
(y=y1 ∨ y=y2 ∨ y=y3)) →
study’(x, y)]]]

There are two students (Mary and Bill) and, for each of them, four theories (TrM1, TrM2, TrM3,
TrM4). Each student studies three theorems for each of the theories associated with her/him.

7
ThM21
ThM22
ThM23
ThB13
ThB12
ThB11
TrB1
ThM43
ThM42
ThM41
ThM13
ThM12
ThM11
ThM33
ThM32
ThM31
TrM4
TrM3
TrM2
TrM1
CONTEXT

x1

x2
[x1≠x2 ∧
s
tud
y

f
ou
r

three
two

x
[(x = x1 ∨ x = x2) → student’(x)] ∧

x
[(x = x1 ∨ x = x2) →

z1

z2

z3

z4
[z1≠z2≠z3≠z4 ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) → theory’(z)] ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) →
M
ar
y

y1

y2

y3
[y1≠y2≠y3 ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) →
(theorem’(y) ∧ belong’(y, z))] ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) →
0 010 00
study’(x, y)]]]]]]

B
il
l

2
2-3-4
24/08/2004

8
CONTEXT
There are three theorems (Th1, Th2 and Th3) and four theories (Tr1, Tr2, Tr3, Tr4). The theorems
belong to all of the theories, and for each theorem there are two students, possibly different, who
study it.
s
tud
y

f
ou
r

three
two

y1

y2

y3

z1

z2

z3

z4
[y1≠y2≠y3 ∧ z1≠z2≠z3≠z4 ∧
Th1
Th2
Th3
Tr1
Tr2
Tr3
Tr4
M
ar
y

B
il
l

A
dam
E
va
H
arr
y

Susan

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) → theory’(z)] ∧

y

z
[((y=y1 ∨ y=y2 ∨ y=y3) ∧
(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4)) →
(theorem’(y) ∧ belong’(y, z))] ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) →

x1

x2
[x1≠x2 ∧
01 0 0

x
[(x = x1 ∨ x = x2) → student’(x)] ∧

x
[(x = x1 ∨ x = x2) → study’(x, y)]]]]

There are three theorems (Th1, Th2 and Th3). Each of them belongs to four theories, possibly
different (Tr11, Tr12, Tr13, Tr14 and Tr21, Tr22, Tr23, Tr24). Moreover, for each theorem there are
two students, possibly different, who study it.
9
CONTEXT
s
tud
y

f
ou
r

three
two

Th1
Th2
Th3
Tr21
Tr22
Tr23
Tr24
A
dam
E
va
H
arr
y

S
usan
B
il
l

M
ar
y

Tr33
Tr34
Tr31
T
r32
Tr11
Tr12
Tr13
Tr14

y1

y2

y3
[y1≠y2≠y3 ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) →
(theorem’(y) ∧

z1

z2

z3

z4
[z1≠z2≠z3≠z4 ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) → theory’(z)] ∧
01 0 01

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) →
belong’(y,z)]])] ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) →

x1

x2
[x1≠x2 ∧

x
[(x = x1 ∨ x = x2) → student’(x)] ∧

x
[(x = x1 ∨ x = x2) → study’(x,y)]]]]

There are four theories (Tr1, Tr2, Tr3, Tr4). For each of them there are three, possibly different
theorems (Th11, Th12, Th13, Th21, Th22, Th23 and Th31, Th32, Th33). For each theorem, there are
two, possibly different, students who study them.
10

Th21
Th22
Th23
Tr1
Tr2
Tr3
Tr4
M
ike
H
elen
E
va
A
dam
Susan
H
arr
y

M
ar
y

B
il
l
Th43
Th42
Th41
Th13
Th12
Th11
Th33
Th32
Th31
CONTEXT

z1

z2

z3

z4
[z1≠z2≠z3≠z4 ∧
s
tud
y

f
ou
r

three
two

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) → theory’(z)] ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) →

y1

y2

y3
[y1≠y2≠y3 ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) →
(theorem’(y) ∧ belong’(y, z))] ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) →

x1

x2
[x1≠x2 ∧

x
[(x = x1 ∨ x = x2) → student’(x)] ∧

x
[(x = x1 ∨ x = x2) → study’(x, y)]]]]

01 010 0

3
2-3-4
24/08/2004

There are four theories (Tr1, Tr2, Tr3, Tr4). For each of them there are two, possibly different
students (who chose them). For each student, there are three, possibly different, theorems who are
studied by that student.
11
CONTEXT
s
tud
y

f
ou
r

three
two
Tr1
Tr3
Tr4
ThH3
ThH2
ThH1
H
arr
y

Susan
Th
S
3
Th
S
2
Th
S
1
Tr2
ThB3
ThB2
ThB1
M
ike
H
elen
A
dam
E
va
B
il
l

M
ar
y

ThK3
ThK2
ThK1
ThM3
ThM2
ThM1

z1

z2

z3

z4
[z1≠z2≠z3≠z4 ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) → theory’(z)] ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) →

x1

x2
[x1≠x2 ∧

x
[(x = x1 ∨ x = x2) → student’(x)] ∧
010 00 0

x
[(x = x1 ∨ x = x2) →

y1

y2

y3
[y1≠y2≠y3 ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) →
(theorem’(y) ∧ belong’(y, z))] ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) →
study’(x, y)]]]]]

There are four theories (Tr1, Tr2, Tr3, Tr4). For each of them there are two, possibly different,
students (who chose them) and three, possibly different, theorems (Th11, Th12, Th13, Th21, Th22,
Th23 and Th31, Th32, Th33). Each student who chose a given theory studies all the theorems of that
theory.
12
CONTEXT
Th21
Th22
Th23
Tr1
Tr2
Tr3
Tr4
H
elen
A
dam
E
va
H
arr
y

Susan
B
il
l

M
ar
y

Th43
Th42
Th41
Th13
Th12
Th11
Th33
Th32
Th31

z1

z2

z3

z4
[z1≠z2≠z3≠z4 ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) → theory’(z)] ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) →
s
tud
y

f
ou
r

three
two

x1

x2

y1

y2

y3
[x1≠x2 ∧ y1≠y2≠y3 ∧

x
[(x=x1 ∨ x=x2) → student’(x)] ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) →
(theorem’(y) ∧ belong’(y, z))] ∧

x

y
[((x=x1 ∨ x=x2) ∧
010 010 0
(y=y1 ∨ y=y2 ∨ y=y3)) →
study’(x, y)]]]

M
ike

There are two students and four theories (Tr1, Tr2, Tr3, Tr4). Each student chooses three theorems
form each theory and studies them.

13
CONTEXT

x1

x2

z1

z2

z3

z4
[z1≠z2≠z3≠z4 ∧ x1≠x2 ∧

x
[(x=x1 ∨ x=x2) → student’(x)] ∧

z
[(z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4) → theory’(z)] ∧
s
tud
y

f
ou
r

three
two
ThM21
ThM22
ThM23
M
ar
y

B
il
l

ThB43
ThB41
ThB42
ThM43
ThM41
ThM42
ThM11
ThM12
ThM13
ThM31
ThM32
ThM33
Tr4
Tr3
Tr2
Tr1

x

z
[((x=x1 ∨ x=x2) ∧ (z=z1 ∨ z=z2 ∨ z=z3 ∨ z=z4)) →

y1

y2

y3
[y1≠y2≠y3 ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) →
(theorem’(y) ∧ belong’(y, z))] ∧

y
[(y=y1 ∨ y=y2 ∨ y=y3) →
study’(x, y)]]]]

010 00+010 0

4